What is the Equation of Director Circle of Hyperbola ?

Solution :

The locus of the intersection of tangents which are at right angles is known as director circle of the hyperbola. The equation to the director circle is :

$x^2+y^2$ = $a^2-b^2$

If $b^2$ < $a^2$, this circle is real ; If $b^2$ = $a^2$ the radius of the circle is zero & it reduces to a point circle at the origin. In this case the center is the only point from which the tangents at right angles can be drawn to the curve.

If $b^2$ > $a^2$, the radius of the circle is imaginary, so that there is no such circle & so no tangents at right angle can be drawn to the curve.